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cosx/3
Plotting a function graph/ cosx/3

Graphing y = cosx/3

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       cos(x)
f(x) = ------
         3
f(x)=cos⁡(x)3f{\left(x \right)} = \frac{\cos{\left(x \right)}}{3} 
f = cos(x)/3
The graph of the function
0-50-40-30-20-101020304050607080900.5-0.5
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
cos⁡(x)3=0\frac{\cos{\left(x \right)}}{3} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=48.6946861306418x_{1} = 48.6946861306418
x2=−54.9778714378214x_{2} = -54.9778714378214
x3=92.6769832808989x_{3} = 92.6769832808989
x4=−2266.65909956504x_{4} = -2266.65909956504
x5=7.85398163397448x_{5} = 7.85398163397448
x6=10.9955742875643x_{6} = 10.9955742875643
x7=1.5707963267949x_{7} = 1.5707963267949
x8=−67.5442420521806x_{8} = -67.5442420521806
x9=26.7035375555132x_{9} = 26.7035375555132
x10=−58.1194640914112x_{10} = -58.1194640914112
x11=−45.553093477052x_{11} = -45.553093477052
x12=−36.1283155162826x_{12} = -36.1283155162826
x13=−4.71238898038469x_{13} = -4.71238898038469
x14=64.4026493985908x_{14} = 64.4026493985908
x15=−95.8185759344887x_{15} = -95.8185759344887
x16=−7.85398163397448x_{16} = -7.85398163397448
x17=−20.4203522483337x_{17} = -20.4203522483337
x18=45.553093477052x_{18} = 45.553093477052
x19=89.5353906273091x_{19} = 89.5353906273091
x20=73.8274273593601x_{20} = 73.8274273593601
x21=61.261056745001x_{21} = 61.261056745001
x22=−76.9690200129499x_{22} = -76.9690200129499
x23=70.6858347057703x_{23} = 70.6858347057703
x24=−89.5353906273091x_{24} = -89.5353906273091
x25=67.5442420521806x_{25} = 67.5442420521806
x26=−48.6946861306418x_{26} = -48.6946861306418
x27=4.71238898038469x_{27} = 4.71238898038469
x28=36.1283155162826x_{28} = 36.1283155162826
x29=−70.6858347057703x_{29} = -70.6858347057703
x30=−42.4115008234622x_{30} = -42.4115008234622
x31=76.9690200129499x_{31} = 76.9690200129499
x32=−51.8362787842316x_{32} = -51.8362787842316
x33=−39.2699081698724x_{33} = -39.2699081698724
x34=54.9778714378214x_{34} = 54.9778714378214
x35=−80.1106126665397x_{35} = -80.1106126665397
x36=−64.4026493985908x_{36} = -64.4026493985908
x37=42.4115008234622x_{37} = 42.4115008234622
x38=−10.9955742875643x_{38} = -10.9955742875643
x39=39.2699081698724x_{39} = 39.2699081698724
x40=−61.261056745001x_{40} = -61.261056745001
x41=20.4203522483337x_{41} = 20.4203522483337
x42=−387.986692718339x_{42} = -387.986692718339
x43=58.1194640914112x_{43} = 58.1194640914112
x44=−14.1371669411541x_{44} = -14.1371669411541
x45=51.8362787842316x_{45} = 51.8362787842316
x46=95.8185759344887x_{46} = 95.8185759344887
x47=32.9867228626928x_{47} = 32.9867228626928
x48=98.9601685880785x_{48} = 98.9601685880785
x49=−26.7035375555132x_{49} = -26.7035375555132
x50=29.845130209103x_{50} = 29.845130209103
x51=−29.845130209103x_{51} = -29.845130209103
x52=−86.3937979737193x_{52} = -86.3937979737193
x53=−32.9867228626928x_{53} = -32.9867228626928
x54=80.1106126665397x_{54} = 80.1106126665397
x55=−83.2522053201295x_{55} = -83.2522053201295
x56=86.3937979737193x_{56} = 86.3937979737193
x57=23.5619449019235x_{57} = 23.5619449019235
x58=−1.5707963267949x_{58} = -1.5707963267949
x59=−3623.82712591583x_{59} = -3623.82712591583
x60=−17.2787595947439x_{60} = -17.2787595947439
x61=14.1371669411541x_{61} = 14.1371669411541
x62=−98.9601685880785x_{62} = -98.9601685880785
x63=−23.5619449019235x_{63} = -23.5619449019235
x64=−73.8274273593601x_{64} = -73.8274273593601
x65=17.2787595947439x_{65} = 17.2787595947439
x66=83.2522053201295x_{66} = 83.2522053201295
x67=−92.6769832808989x_{67} = -92.6769832808989
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)/3.
cos⁡(0)3\frac{\cos{\left(0 \right)}}{3}
The result:
f(0)=13f{\left(0 \right)} = \frac{1}{3}
The point:
(0, 1/3)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−sin⁡(x)3=0- \frac{\sin{\left(x \right)}}{3} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
    1 
(0, -)
    3 

(pi, -1/3)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=πx_{1} = \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Increasing at intervals
[0,π]\left[0, \pi\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−cos⁡(x)3=0- \frac{\cos{\left(x \right)}}{3} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(cos⁡(x)3)=⟨−13,13⟩\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{3}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−13,13⟩y = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle
lim⁡x→∞(cos⁡(x)3)=⟨−13,13⟩\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{3}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−13,13⟩y = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)/3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)3x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{3 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)3x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{3 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
cos⁡(x)3=cos⁡(x)3\frac{\cos{\left(x \right)}}{3} = \frac{\cos{\left(x \right)}}{3}
- Yes
cos⁡(x)3=−cos⁡(x)3\frac{\cos{\left(x \right)}}{3} = - \frac{\cos{\left(x \right)}}{3}
- No
so, the function
is
even
The graph
Graphing y = cosx/3
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